European Institute for Statistics, Probability, Stochastic Operations Research
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May 11-12, 2011


Workshop on

Random Graphs and the Brain






The meeting combines researchers from random graphs and complex networks, and from neuroscience. The aim is to obtain a better understanding of how random graphs, or probability theory in general, can play a role in neuroscience.
This workshop fits within the framework of the SAM



Remco van der Hofstad Eindhoven University of Technology/EURANDOM


Frank Redig Radboud University Nijmegen


Wioletta Ruszel Eindhoven University of Technology/Radboud University Nijmegen





Registration is obligatory for all participants (organizers and speakers too!).

Registration fee is 75 euros, to be paid by all participants (excluding speakers and TU/e personnel)

Please follow the link to the online registration. (closed)

(For the registration form you will be redirected to the website of the Eindhoven University of Technology)



Invited speakers

Deijfen, Mia Stockholm University
Dommers, Sander Eindhoven University of Technology
Gómez, Vicenç Radboud University Nijmegen
Memmesheimer, Raoul-Martin Radboud University Nijmegen
Roudi, Yasser Kavli Insitute, NTNU
Straaten, Ilse van VU Medical Center
Turova, Tatyana Lund University
Van Mieghem, Piet Delft University of Technology



Wednesday May 11

09.30 - 09.45 Opening  
09.45 - 10.30 Piet Van Mieghem Measuring randomness and structure in complex networks
10.45 - 11.30 Raoul-Martin Memmesheimer Irregular spiking activity in random neural networks
11.45 - 12.30 Yasser Roudi Mean-fields theory for non-equilibrium network reconstruction and applications to neural data
12.30 - 14.30 Lunch  
14.30 - 15.15 Vicenç Gómez Self-organization using synaptic plasticity
15.30 - 16.15 Sander Dommers Ising models on power-law random graphs
16.15 - 17.45 Discussion with some drinks  
17.45 - .... Diner  


Thursday May 12

09.30 - 10.15 Mia Deijfen Scale-free percolation
10.30 - 11.15 Ilse Straaten How to capture functional connectivity in the Brain
11.30 - 12.15 Tatyana Turova The emergence of connectivity in neuronal networks:
from bootstrap percolation to auto-associative memory
12.15 - 13.15 Lunch  
13.15 - 15.00 Open discussion What role does probability theory play in neuroscience and what are challenging new problems?
15.00 - ......... Discussion with some drinks  



Mia Deijfen

Scale-free percolation

I will describe a model for inhomogeneous long-range percolation on Z^d with potential applications in neuroscience. Each vertex is independently assigned a non-negative random weight and the probability that there is an edge between two given vertices is then determined by a certain function of their weights and of the distance between them. The results concern the degree distribution in the resulting graph, the percolation properties of the graph and the graph distance between remote pairs of vertices. The model interpolates between long-range percolation and inhomogeneous random graphs, and is shown to inherit the interesting features of both these model classes.


Sander Dommers

Ising models on power-law random graphs

We study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent $\tau > 2$), for which the random graph has a tree-like structure. For this, we adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($\tau > 3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy.
(Joint work with C. Giardinŕ and R. van der Hofstad).


Vicenç Gómez

Self-organization using synaptic plasticity

Large networks of spiking neurons show abrupt changes in their collective dynamics resembling phase transitions studied in statistical physics. An example of this phenomenon is the transition from irregular, noise-driven dynamics to regular, self-sustained behavior observed in networks of integrate-and-fire neurons as the interaction strength between the neurons increases. In this work we show how a network of spiking neurons is able to self-organize towards a critical state for which the range of possible inter-spike-intervals (dynamic range) is maximized. Self-organization occurs via synaptic dynamics that we analytically derive. The resulting plasticity rule is defined locally so that global homeostasis near the critical state is achieved by local regulation of individual synapses.


Raoul-Martin Memmesheimer

Irregular spiking activity in random neural networks

Mean field theory for infinite sparse networks of spiking neurons shows that a chaotic balanced state of highly irregular activity arises under a variety of conditions. The state is considered a model for the ground state of cortical activity. We analytically investigate the irregular dynamics of a balanced state in finite random networks keeping track of all individual spike times and the identities of individual neurons. For delayed, purely inhibitory interactions, we show that the dynamics is not chaotic but in fact stable. Moreover, we demonstrate that after long transients the dynamics converges towards periodic orbits and that every generic periodic orbit of these dynamical systems is stable. These results indicate that chaotic and stable dynamics are equally capable of generating the irregular neuronal activity. More generally, chaos apparently is not essential for generating high irregularity of balanced activity, and we suggest that a mechanism different from chaos and stochasticity significantly contributes to irregular activity in cortical circuits.


Yasser Roudi

Mean-fields theory for non-equilibrium network reconstruction and applications to neural data

With the advancement of multi-electrode recording, an important and interesting question is arising: can we infer interactions between neurons from the simultaneous observation of spike trains from many neurons? In this talk, I first describe a toy version of this problem:
how can we infer interactions of an Ising model from observing its state samples? I will start by a short review of how this could be done for equilibrium systems and then study how Dynamical Mean-Field (naive mean field and TAP) theory can be developed for a non-equilibrium Ising model and exploited for inferring the network connectivity. Finally, I will describe how all this can be used to for inferring connectivity from neural multi-electrode recordings.


Ilse Straaten

How to capture functional connectivity in the Brain

How can different, specialized brain areas communicate with each other in order to come to a high level of functioning, such as consciousness, attention, and various cognitive functions? The brain is complex. Proper functioning requires both the segregation as well as the integration of information processing. One of the central questions in neuroscience is how this integration takes place. Neurons have intrinsic electrical firing properties and they tend to synchronize this activity with great precision when they are connected. Synchronization of neuronal activity is therefore a sign of functional connectivity. Time series of brain activity can be measured with EEG, MEG, and fMRI in various brain areas and several methods exist for determining the strength of correlation between these regional time series. One of the nonlinear methods, the phase lag index (PLI), will be discussed in more detail. With a large number of measurement points on the skull (the nodes in the network), the result is a large matrix of correlation values. When modeled as graphs, concepts and analytical measures derived from modern network theory can be applied. As it turns out, healthy brains of subjects in eyes-closed resting-state show a small-world modular structure, characterized by various properties such as sparse connectivity, high clustering, short path-lengths between any two network points, skewed degree distributions, presence of highly connected and central hubs and a hierarchical architecture with modules and sub-networks. Optimal brain function requires efficient underlying anatomical networks. The MRI technique of diffusion tensor imaging (DTI) is capable of visualizing the cerebral anatomical wiring. A small-world organization was also found for these structural networks. Although functional and anatomical networks are not overlapping completely, analogies can be found. More recently, functional changes in several neurological and psychiatrical disorders have been studied. Some of these will be discussed among which Alzheimer’s disease and brain tumors. In addition, modeling can be used to understand the rules underlying brain network development and changes in several types of disease. All together, the field of network science has provided a complete new view on neuronal function and disease.


Tatyana Turova

The emergence of connectivity in neuronal networks: from bootstrap percolation to auto-associative memory

We consider some examples of models for neural networks (accepted by the neuro-physiological
community) where the underlying graph of connections is random. We show that an architecture with almost equal numbers of in- and out-connections can emerge naturally in a class of networks with random connections, and an exact example of such a network is provided. It is proved that the set of parameters under which the propagation of excitation has a constant speed is rather small within the whole space of possible parameters.
We shall discuss which properties of random graphs are relevant for the neuro - physiological studies. In particular, we argue that the threshold of connectivity for an auto-associative memory in a Hopfield model with a random connection matrix coincides with the threshold for bootstrap percolation on the corresponding random graph.


Piet Van Mieghem

Measuring randomness and structure in complex networks

What is the relations between the properties of a complex network and the amount of randomness and structure in that complex network? It is conjectured that the randomness of a brain network reduces from birth with age and that the process of learning creates structure in the brain. We will briefly summarize the techniques to measure randomness and propose new methods for measuring randomness and structure in networks. The method will be applied to a set of Altzheimer patients and a set of "healthy controls".
This work is in collaboration with the group of Professor Stam (VUMc, Amsterdam).



Practical information

Conference Location
The workshop location is EURANDOM,  Den Dolech 2, 5612 AZ Eindhoven, Laplace Building, 1st floor, LG 1.105.

EURANDOM is located on the campus of Eindhoven University of Technology, in the 'Laplacegebouw' building' (LG on the map). The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm

For more information please contact Mrs. Patty Koorn,
Workshop officer of  EURANDOM


Sponsored by:

tochastics-Theoretical and Applied Research


Last updated 14-07-11,
by PK